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Hawkins compatibility conditions on the tangent bundle of a Poisson-Lie group. (English) Zbl 1393.53084

Summary: Let \(G\) be a Poisson-Lie group equipped with a left invariant Riemannian metric compatible with the Poisson structure on \(G\). There are many ways to lift the Poisson structure and the metric to the tangent bundle \(TG\) of \(G\). In this paper, we study in different cases the compatibility between the lifted Poisson structure and the lifted metric on \(TG\).

MSC:

53D17 Poisson manifolds; Poisson groupoids and algebroids
58B20 Riemannian, Finsler and other geometric structures on infinite-dimensional manifolds
70G65 Symmetries, Lie group and Lie algebra methods for problems in mechanics
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References:

[1] Bahayou, A., and M. Boucetta, Metacurvature of Riemannian Poisson-Lie groups, J. of Lie Theory 19 (2009), 439-462. · Zbl 1186.58006
[2] Boucetta, M., Compatibilit´e des structures pseudo-riemanniennes et des struc tures de Poisson, C. R. Acad. Sci. Paris, S´erie I, 333 (2001), 763-768. · Zbl 1009.53057
[3] —, Poisson manifolds with compatible pseudo-metric and pseudo-Riemannian Lie algebras, Diff. Geometry and its Applications 20 (2004), 279-291. · Zbl 1061.53058
[4] —, On the Riemann-Lie Algebras and Riemann-Poisson Lie groups, J. of Lie Theory 15 (2005), 183-195. · Zbl 1077.53065
[5] Boumaiza, M., and N. Zalaani, Poisson-Lie structure on the tangent bundle of a Poisson-Lie group, and Poisson action lifting, J. of Geometry and Symmetry in Physics 4 (2005), 1-17. 524 Aloui and Zaalani · Zbl 1132.53315
[6] Cheeger, J., and D. Gromoll, On the structure of complete manifolds of nonnegative curvature, Ann. of Math. 96 (1972), 413-443. · Zbl 0246.53049
[7] Fernandes, R. L., Connections in Poisson geometry I: Holonomy and invari ants, J. Diff. Geom. 54 (2000), 303-365. · Zbl 1036.53060
[8] Hawkins, E., Noncommutative rigidity, Com.Math.Phys. 246 (2004), 211- 235. · Zbl 1055.58001
[9] —, The structure of noncommutative deformations, J. Diff. Geom. 77 (2007), 385-424. · Zbl 1130.53062
[10] Milnor, J., Curvature of left invariant metrics on Lie groups, Adv. in Math. 21 (1976), 293-329. · Zbl 0341.53030
[11] Sanchez de Alvarez, G., Poisson brackets and dynamics, Dynamical systems, Santiago (1990), 230-249. · Zbl 0791.58042
[12] Sasaki, S., On the differential geometry of tangent bundles of Riemannian manifolds, Tohoku Math. J. 10 (1958), 338-354. · Zbl 0086.15003
[13] Vaisman, I., “Lectures on the Geometry of Poisson Manifolds,” Progr. in Math. 118 Birkh¨auser, Berlin, (1994). · Zbl 0810.53019
[14] Weinstein, A., Some remarks on · Zbl 0653.58012
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